Trigonometric Equations

What Is a Trigonometric Equation?

Trigonometric equations are equations where the unknown variable, x, serves as the argument of one or more trigonometric functions (such as sine, cosine, or tangent).

A Practical Example

Here’s an example of a trigonometric equation:

$$ x^2 + \sin(x) -1 $$

This qualifies as a trigonometric equation because the unknown variable x appears as the argument of the sine function.

Example 2

The following, however, is NOT a trigonometric equation:

$$ x^2 + \sin(\frac{\pi}{2}) -1 $$

because the unknown x does not serve as the argument of any trigonometric function here.

Note: In this case, the argument of the sine function is pi over two, which is a constant, not an expression containing x.

Types of Trigonometric Equations

Trigonometric equations come in different forms:

• Elementary Trigonometric Equations
  These equations involve the unknown x as the argument of a trigonometric function. Examples of elementary trigonometric equations include $$ \sin x = a $$ $$ \cos x = b $$ $$ \sin \alpha = \cos \beta $$ $$ \tan \alpha = -\tan \beta $$ where a and b are real numbers.
• Linear Equations in Sine and Cosine
  These trigonometric equations can be expressed in the form $$ a \cdot \sin x + b \cos x + c = 0 $$ where a, b, and c are real numbers (the coefficients of the linear equation), and neither a nor b is zero.
• Second-Degree Equations in Sine and Cosine
  These are trigonometric equations that can be written in the form $$ a \cdot \sin^2 x + b \sin x \cos x + c \cdot \cos^2 x = 0 $$ where a, b, and c are real numbers (the equation’s coefficients), and neither a nor b is zero.

And so forth.